Math: Basic Tutorials : Fractions

Understanding Fractions

Fractions are essential for performing accurate calculations in healthcare. A fraction represents a part of a whole and is expressed as a numerator (the part) over a denominator (the whole). Understanding fractions and their different forms—proper, improper, and mixed—helps you handle a variety of clinical scenarios with ease.

This section will explore the different forms of fractions and how to switch between them, making it easier to perform operations in different contexts. Additionally, we’ll cover arithmetic operations with fractions, including addition, subtraction, multiplication, and division, all of which are critical for accurate medication calculations and other healthcare tasks.

Understanding Fractions Video transcript - RTF

Proper Fractions

A proper fraction is a fraction whose numerator is smaller than its denominator. For example:

$\frac{1}{2} or \frac{4}{5} or \frac{3}{7}$

Example:
Consider the proper fraction: $\frac{4}{5}$ It can be visualized as:

Example:

Consider the proper fraction: $\frac{4}{5}$

It can be visualized as:

pie chart with 4 out of 5 sections filled

Improper Fractions

An improper fraction is a fraction whose numerator is greater than its denominator. For example:

$\frac{3}{2} or \frac{7}{4} or \frac{9}{5}$

Example:
Consider the improper fraction: $\frac{7}{4}$ It can be visualized as:

Example:

Consider the improper fraction: $\frac{7}{4}$

It can be visualized as:

pie chart with 4 out of 4 sections filled

pie chart with 3 out of 4 sections filled

Mixed Fractions

A mixed fraction or mixed number is a combination of a whole number and a proper fraction. For example:

$3 \frac{1}{2} or 5 \frac{2}{5} or 1 \frac{3}{4}$

Example:
The mixed fraction $1 \frac{3}{4}$ means there is one whole part plus $\frac{3}{4}$ of a whole part.
In fact, $1 \frac{3}{4} = 1 + \frac{3}{4}$ and can be visualized as:

Example:

The mixed fraction $1 \frac{3}{4}$ means there is one whole part plus $\frac{3}{4}$ of a whole part.

In fact, $1 \frac{3}{4} = 1 + \frac{3}{4}$

and can be visualized as:

pie chart whole

pie chart with 3 out of 4 sections filled

Note: the shaded region in the circles representing $\frac{7}{4}$ from 'improper fractions' and $1 \frac{3}{4}$ are the same. In fact, $1 \frac{3}{4} = \frac{7}{4}$

They are different ways of representing the same thing!

Switch from Mixed to Improper Fractions

To switch from mixed to improper:

Multiply the whole number by the denominator of the proper fraction
Add to the numerator of the proper fraction.

Example 1:
$4 \frac{2}{3} = \frac{4 \times 3 + 2}{3} = \frac{12 + 2}{3} = \frac{14}{2}$
Example 2:
$5 \frac{1}{4} = \frac{5 \times 4 + 1}{4} = \frac{20 + 1}{4} = \frac{21}{4}$
Example 3:
$2 \frac{3}{7} = \frac{2 \times 7 + 3}{7} = \frac{14 + 3}{7} = \frac{17}{7}$

Switch from Improper to Mixed Fractions

To switch from improper to mixed:

Use long division to divide the denominator into the numerator.
We will use the Quotient and Remainder to write our Mixed Fraction.

The improper fraction in mixed form is: $Quotient \frac{Remainder}{Denominator}$

Example:
Suppose we want to change $\frac{21}{4}$ to a mixed fraction.
Solution:	21 divided by 4 is 5 with a remainder of 1. So: $\frac{21}{4} = 5 \frac{1}{4}$

Adding and Subtracting Fractions

To add/subtract fractions:

Make sure the denominators are the same.**
Add/Subtract the numerators and put that answer over the denominator.
Simplify the fraction if necessary.
- This is done by dividing numerator and denominator by their Greatest Common Factor (GCF).

**If they are not, this can be done by rewriting one/both of the fractions by multiplying both numerator and denominator by the same value. For example:

$\frac{1}{2}$ is the same as $\frac{2}{4}$ because $\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4}$

Note: For simplicity, we aim to multiply both fractions separately to ensure each denominator becomes the least common multiple (LCM) of the two original denominators.

Click on the titles below to view each example.

Example 1

Example 1:
Suppose we want to add: $\frac{3}{12} + \frac{5}{12}$
The denominators are the same, so we add the numerators and simplify:	$\frac{3}{12} + \frac{5}{12} = \frac{8}{12}$ $= \frac{8 \div 4}{12 \div 4}$
Solution:	$= \frac{3}{12}$
The above calculation can be visualized as:

Example 2

Example 2:
Suppose we want subtract: $\frac{2}{5} - \frac{1}{4}$
The denominators are not the same so we must rewrite one/both of the fractions. The least common multiple of 5 and 4 is 20, so we need to multiply each fraction above and below so that their denominator becomes 20.	$\frac{2}{5} = \frac{2 \times 4}{5 \times 4}$ and $\frac{1}{4} = \frac{1 \times 5}{4 \times 5}$ $= \frac{8}{20}$ $= \frac{5}{20}$
Now that the denominators are the same, the fractions can be subtracted as usual:	$\frac{2}{5} - \frac{1}{4} = \frac{8}{20} - \frac{5}{20}$
Solution:	$= \frac{3}{20}$
The above calculation can be visualized as:

Multiplying Fractions

To multiply fractions:

Multiply the numerators together.
Multiply the denominators together.
As always, simplify if necessary.

Note: remember the rule "Top by top, Bottom by bottom".

Example:
Multiply the numerators and the denominators.	$\frac{3}{5} \times \frac{2}{7} = \frac{3 \times 2}{5 \times 7}$
Solution:	$= \frac{6}{35}$

Dividing Fractions

To divide fractions:

Change the division symbol to a multiplication symbol.
Flip the divisor.
- The divisor is the dividing fraction in this case, like $\frac{1}{2}$ in $\frac{3}{4} \div \frac{1}{2}$
Perform the multiplication, and as always, simplify if necessary.

Note: remember the rule "Keep, Change, Flip".

Example:
$\frac{2}{9} \div \frac{3}{4}$
Change the division symbol to a multiplication symbol. Keep the $\frac{2}{9},$ flip the $\frac{3}{4}$ Then multiply top by top and bottom by bottom.	$\frac{2}{9} \div \frac{3}{4} = \frac{2}{9} \times \frac{4}{3}$ $= \frac{2 \times 4}{9 \times 3}$
Solution:	$= \frac{8}{27}$

Example:

$\frac{2}{9} \div \frac{3}{4}$

Change the division symbol to a multiplication symbol.

Keep the $\frac{2}{9},$ flip the $\frac{3}{4}$

Then multiply top by top and bottom by bottom.

$\frac{2}{9} \div \frac{3}{4} = \frac{2}{9} \times \frac{4}{3}$

$= \frac{2 \times 4}{9 \times 3}$

Solution:

$= \frac{8}{27}$

Practice

Additional Resources

Learning Portal Basic Tutorials: Fractions

Last Updated: Jun 5, 2025 4:16 PM
URL: https://tlp-lpa.ca/math-tutorials
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Math: Basic Tutorials

Math: Basic Tutorials : Fractions

Understanding Fractions

Proper Fractions

Improper Fractions

Mixed Fractions

Switch from Mixed to Improper Fractions

Switch from Improper to Mixed Fractions

Adding and Subtracting Fractions

Example 1

Example 2

Multiplying Fractions

Dividing Fractions

Practice

Additional Resources